Page:Scientific Memoirs, Vol. 1 (1837).djvu/469

Rh By making the substitutions in the equation (4) we shall be able to exhibit the result in the following form: The foregoing equation is satisfied by taking

This equation is of the same form with that proposed (4), except only that $$i$$ is replaced by $$i-1$$. If therefore again, in the latter, we put we shall deduce from it another in terms of $$Q_i^{(2)}$$, in which $$i-1$$ will be replaced by $$i-2$$; and by continuing these substitutions we shall finally obtain the equation which is integrable by the known methods, and gives where $$T_i$$ and $$V_i$$ may be considered as two arbitrary functions of $$\theta$$ and $$\psi$$ of the order $$i$$ which satisfy the equation (3).

By adopting this value $$Q_i^{(i)}$$, and by afterwards taking