132 



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

 dimensions, if we write it under the form 



0.a:"4-0.^"-' + O.x"-^-^ 4.0.r'»+'4- 



ax"^ + /3:r"'-i + &c. = 0. 



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

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

 written are zero, the first i of the quantities 



{a'Uo - aFo), {b'Uo -bV,) + {a'U, - aV,), &c. 



may be read simply 



- « . Fo, - 6. Fo - aF„ - c Fo - 6r, - aV^, &c. 



and evidently their office can be supplied by the simple 

 augmentatives themselves 



Fo = 0, F, = 0, V,-0 V,_,z=.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 + /uar + v^r" + . . . . + 0:r'«-'-\ 



and Yr of the form 



l + mx J\ +<a:"-'"-', 



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

 simple equations being required for the determination of the 

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

 ing these latter as known linear functions of no more than 

 {n^— r) unknown quantities left to be determined. 



