There was an air of jaded sullenness in them both, and particularly in the girl: yet, struggling through the dissatisfaction of [Louisa's] face, there was a light with nothing to rest upon, a fire with nothing to burn, a starved imagination keeping life in itself somehow, which brightened its expression. Not with the brightness natural to cheerful youth, but with uncertain, eager, doubtful flashes, which had something painful in them, analogous to the changes on a blind face groping its way. She was a child now, of fifteen or sixteen; but at no distant day would seem to become a woman all at once. (1.3.20-22)
As a child, Louisa is already trapped between what she has been taught to do and what her natural inclinations are. This is the first time we see her, and already the image of fire is applied to her. She has a restless "light" and a "fire" that has no fuel, like a walking bomb.
Mrs. Gradgrind was not a scientific character, and usually dismissed her children to their studies with this general injunction to choose their pursuit. In truth, Mrs. Gradgrind's stock of facts in general was woefully defective; but Mr. Gradgrind in raising her to her high matrimonial position, had been influenced by two reasons. Firstly, she was most satisfactory as a question of figures; and, secondly, she had 'no nonsense' about her. By nonsense he meant fancy; and truly it is probable she was as free from any alloy of that nature, as any human being not arrived at the perfection of an absolute idiot, ever was. (1.4.29-31)
Mrs. Gradgrind is one extreme example of a socially acceptable kind of femininity in the novel. She is entirely submissive to her husband, entirely unable to function without his guidance and assistance. It's saying something that this kind femininity is here described as idiotic (considering Dickens wasn't the world's most progressive guy when it came to women's issues).
'Tom,' said his sister, after silently watching the sparks awhile, 'as I get older, and nearer growing up, I often sit wondering here, and think how unfortunate it is for me that I can't reconcile you to home better than I am able to do. I don't know what other girls know. I can't play to you, or sing to you. I can't talk to you so as to enlighten your mind, for I never see any amusing sights or read any amusing books that it would be a pleasure or a relief to you to talk about, when you are tired.' (1.8.17)
Louisa's confession that she cannot amuse Tom is in stark contrast to Sissy, who we know read to her father to make him feel better after a poor performance. And yet, Sissy's father still abandoned her. Would Tom really have turned out that different even if Louisa could tell him amusing stories?
'I have such unmanageable thoughts,' returned his sister, 'that they will wonder.' (1.8.43)
"Unmanageable" is a key word for the novel. It seems applicable to the repressed feelings of many, many characters. Almost every emotion anyone is carrying around inside is unmanageably, damaging, or destructive in some way.
'No! Don't, please; don't. Let me see thee setten by the bed. Let me see thee, a' so good, and so forgiving. Let me see thee as I see thee when I coom in. I can never see thee better than so. Never, never, never!' He had a violent fit of trembling, and then sunk into his chair. After a time he controlled himself, and, resting with an elbow on one knee, and his head upon that hand, could look towards Rachael. Seen across the dim candle with his moistened eyes, she looked as if she had a glory shining round her head. (1.13.31-32)
Stephen can't allow himself to think about Rachael in a sexualized way (see the moment when he tell Bounderby that she could never be with him outside of marriage). So, he instead transforms her into an angel of purity, another kind of socially acceptable type of femininity. Not sure how fair this is to Rachael herself, though.
[Gradgrind] really liked Sissy too well to have a contempt for her; otherwise he held her calculating powers in such very slight estimation that he must have fallen upon that conclusion. Somehow or other, he had become possessed by an idea that there was something in this girl which could hardly be set forth in a tabular form. Her capacity of definition might be easily stated at a very low figure, her mathematical knowledge at nothing; yet he was not sure that if he had been required, for example, to tick her off into columns in a parliamentary return, he would have quite known how to divide her. (1.14.22)
Sissy is the first woman Gradgrind comes across that challenges his philosophical system. Mrs. Gradgrind just let the system run over her like a truck, and Louisa molded herself to fit it. But Sissy's natural empathy and emotional connection with others can't be untaught.
'Father,' [Louisa] returned, almost scornfully, 'what other proposal can have been made to me? Whom have I seen? Where have I been? What are my heart's experiences? […] What do I know, father,' said Louisa in her quiet manner, 'of tastes and fancies; of aspirations and affections; of all that part of my nature in which such light things might have been nourished? What escape have I had from problems that could be demonstrated, and realities that could be grasped? […] You have been so careful of me, that I never had a child's heart. You have trained me so well, that I never dreamed a child's dream. You have dealt so wisely with me, father, from my cradle to this hour, that I never had a child's belief or a child's fear.' Mr. Gradgrind was quite moved by his success, and by this testimony to it. 'My dear Louisa,' said he, 'you abundantly repay my care. Kiss me, my dear girl.' (1.15.46-51)
Louisa is talking about the fact that she has had no other marriage proposals aside from Bounderby's. So, it seems odd that she ends up talking about the way she differed from other children and their emotional lives. What's the connection between "a child's belief" or "a child's fear" and the fact that she doesn't know any people who could have fallen in love with her?
Mrs. Sparsit was conscious that by coming in the evening-tide among the desks and writing implements, she shed a feminine, not to say also aristocratic, grace upon the office. Seated, with her needlework or netting apparatus, at the window, she had a self-laudatory sense of correcting, by her ladylike deportment, the rude business aspect of the place. With this impression of her interesting character upon her, Mrs. Sparsit considered herself, in some sort, the Bank Fairy. The towns-people who, in their passing and repassing, saw her there, regarded her as the Bank Dragon keeping watch over the treasures of the mine. (2.1.9)
Mrs. Sparsit is the only woman in the novel who very self-consciously acts out what she believes to be socially acceptable feminine behavior. She has no idea what the bank does, or what the clerks do (she looks at the scraps of their writing and can't interpret them). But even when she is alone, like here, she makes sure to move around the bank in a "feminine" manner, congratulating herself on looking like "the Bank Fairy."
there presently entered to them the most remarkable girl Mr. James Harthouse had ever seen. She was so constrained, and yet so careless; so reserved, and yet so watchful; so cold and proud, and yet so sensitively ashamed of her husband's braggart humility — from which she shrunk as if every example of it were a cut or a blow; that it was quite a new sensation to observe her. In face she was no less remarkable than in manner. Her features were handsome, but their natural play was so locked up, that it seemed impossible to guess at their genuine expression. Utterly indifferent, perfectly self-reliant, never at a loss, and yet never at her ease, with her figure in company with them there, and her mind apparently quite alone — it was of no use 'going in' yet awhile to comprehend this girl, for she baffled all penetration. From the mistress of the house, the visitor glanced to the house itself. There was no mute sign of a woman in the room. No graceful little adornment, no fanciful little device, however trivial, anywhere expressed her influence. Cheerless and comfortless, boastfully and doggedly rich, there the room stared at its present occupants, unsoftened and unrelieved by the least trace of any womanly occupation. (2.2.22-23)
In the nineteenth century, the domestic space (a.k.a. the home) was supposed to be the woman's sphere, where she could express herself and use her power to create a safe, comfortable space. It's obviously significant that Louisa's house with Bounderby is just as blank as her face and body language, none of which display any significantly female or feminine traits beyond good looks.
Mr. Harthouse drew a long breath; and, if ever man found himself in the position of not knowing what to say, made the discovery beyond all question that he was so circumstanced. The childlike ingenuousness with which his visitor spoke, [Sissy's] modest fearlessness, her truthfulness which put all artifice aside, her entire forgetfulness of herself in her earnest quiet holding to the object with which she had come; all this, together with her reliance on his easily given promise — which in itself shamed him — presented something in which he was so inexperienced, and against which he knew any of his usual weapons would fall so powerless, that not a word could he rally to his relief. (3.2.38)
This kind of woman, and this kind of female power, is a recurring favorite for Dickens – and actually the way that the Victorian age liked to see its women. Sissy is so morally pure that she somehow can muster power over a man who is socio-economically her superior. The problem, of course, is that this kind of moral purity is only attainable for a character in a novel. For real women, Sissy is not a realistic example, nor a fair goal.