The poem "The Mother's Smile" by William Butler Yeats describes the complex emotions and thoughts of a daughter who is about to leave her mother for her own responsibilities. The daughter's smile signifies her assurance to her mother and her helplessness inside. Here's a detailed explanation:

Assurance to mother:
- The daughter's smile is described as "brightening her face" in the poem. This suggests that she is trying to reassure her mother that everything will be alright.
- The daughter says "I must depart / All that I am, all that I have been." This shows that she is aware of the impact her departure will have on her mother, and she wants to make sure that her mother knows that she is still the same person, even though she is leaving.
- The daughter also says "But I will come again, and I will bring / New happiness and youthfulness." This further reinforces her assurance to her mother that she will return and bring joy and vitality with her.

Helplessness inside:
- Despite her attempts to reassure her mother, the daughter is feeling helpless inside. She says "But I cannot weep, / For all my body's sleepless agony." This suggests that she is struggling to contain her own emotions, and her smile is a way of masking her true feelings.
- The daughter also says "And yet I cannot stay, / For all my heart's allegiance is with you." This shows that she is torn between her responsibilities and her love for her mother. She wants to stay with her mother, but she knows she must leave to fulfill her duties.

In conclusion, the poet's smile in the poem "The Mother's Smile" signifies her assurance to her mother and her helplessness inside. She is trying to reassure her mother that everything will be alright, but she is struggling to contain her own emotions. The poem highlights the complex emotions and thoughts that arise when we have to leave our loved ones for our own responsibilities.