The combination of symptoms described points towards a specific homeopathic remedy. Let's analyze the key characteristics:
Light hair and skin: This physical constitution is often associated with certain homeopathic types.
Irritable, quarrelsome disposition; easily offended at trifles; vexed at everything:** This describes a particular mental-emotional state.
Skin of hands rough, cracked; tips of fingers rough, cracked, fissured, every winter:** This suggests a chronic skin condition that worsens in winter.
Tenderness of the feet which are bathed in foul-smelling sweat:** This indicates foot problems with a characteristic offensive odor.
Considering these symptoms together, Petroleum is a well-known homeopathic remedy that fits this picture closely. Keynotes for *Petroleum* include:
Skin problems that are worse in winter, with roughness, cracking, and fissures, especially on the hands and fingertips.
A characteristic of offensive foot sweat.
An irritable and easily offended disposition.
While other remedies might share some of these symptoms, the specific combination is most indicative of *Petroleum*.
Let's briefly consider why the other options are less likely:
Silicea: *Silicea* individuals often have a delicate constitution, can be chilly, and may have offensive foot sweat, but the skin symptoms and the specific irritability are less typical.
Graphites: *Graphites* is known for skin conditions with oozing, stickiness, and cracking, often with a history of eczema. The mental disposition is usually more anxious or hesitant rather than quarrelsome.
Sanicula: *Sanicula* is often indicated for children with a specific set of symptoms, including offensive foot sweat that smells like old cheese, and skin problems that might involve crusts or eruptions, but the overall picture doesn't align as closely with the described irritability and winter aggravation of skin.
Therefore, based on the totality of the described symptoms, *Petroleum* is the most likely homeopathic remedy.