370 Mk MURPHY'S THIRD MEMOIR ON THE 



Let S = UaVo + fli Fi . ss + (hV.i.%^ + ... + «x ^^K' + &c. 

 be the series proposed. 



By the inverse method, put a^ = U/{t) . t' from t = to t = 1 . 



Then S = /^/(r) \V, + F.tz + V.t'z' + &c.| 



But V^+ V^k + VJe^, &;c. is the function which V„ generates, and may 

 be represented by ^ {t, k), we have then 



S = X-/(t) . <p{t, T%), from T = to T = 1. 



