133 



The equation V may be conceived to be of n instead of m 

 dimensions, if we write it under the form 



a^™ + j3a;™-i + &c. = 0. 



and we are able to apply the same method as above ; 

 but as the first t of the coefficients in the equation above 

 written are zero, the first i of the quantities 



{a'U, - aVo), {b'U, -bF,)^ {a'U^ - aV,), &c. 



may be read simply 



-a.Vo, -b.Vo- aVu -cV^- bV,- aV^, &c. 



and evidently their office can be supplied by the simple 

 augmentatives themselves 



ro = o, v, = o, v, = o K-, = 0; 



and thus i letters, which otherwise would be irrelevant, fall 

 out of the several derivees. 



The Author then proceeds with remarks upon the gene- 

 ral theory of simple equations, and shows how by virtue of 

 that theory his method contains a solution of the identity 



Xr.U-^Yr.V=Dr; 



where Dr is a derivee of the r'^ degree of U and V, and, 

 accordingly, Xr of the form 



X + fix + vx"" + ....-{- ex'^-'-\ 



and Yr of the form 



and accounts a priori for the fact of not more than (n — r) 

 simple equations being required for the determination of the 

 {m +w — 2r) quantities X, fx, v, &c. /, m, n, &c., by exhibit- 

 ing these latter as known linear functions of no more than 

 (^fi.— r) unknown quantities left to be determined. 



