VOLUMES OF SOLIDS AS BELATED TO TRANSVERSE SECTIONS. 41 



of suitably conditioning all equations beyond the (n - l)th will 

 depend upon exponential relations. Let the index of a, b, etc., 

 in the (n + k)th equation — that is the mfch line or row in (8) — 

 be u; then from (7), (9) and (10) we have, similarly to (6) 



-r<.-— i,W<6»- -L-U. .. + (»«- ')=/(«) = o... (13) 



u+ 1/ u+ Y) u+ 1/ 



as the general form of the equation, in which u is to be so deter- 

 mined as to make its value zero. It is obvious that the values 

 p, q, r etc. are all solutions of this equation : consequently any 

 higher index than that of the rath line or row in equations (8), 

 must be a solution of (13) other than those already to hand. 

 Consequently, 



Prop. (b). If the number of indices in the original function 

 exceed by k, any given number of arbitrary values thereof, k + 1 

 of the indices must be conditioned. 



The condition is that/ (w) = 0, where the function is of the 

 form (13). 



5. Number of indices less than the number of values of the 

 variable diminished by unity. — If on the other hand the number 

 of indices in (10) be m = n-k, then there will be k - 1 too few 

 equations for the determination of the ratios of the whole of the 

 weight-coefficients : i.e., there will in general be a &-fold infinity 

 of solutions, and k - 1 of the ratios or k of the weights may be 

 arbitrarily assumed, or k - 1 suitable conditions may be imposed 

 upon them. Consequently 



Prop. (c). If any given number of arbitrary values of the 

 original function, exceed by k the number of different indices 

 therein, then k - 1 of the weight-coefficient-ratios, or k coefficients 

 must be arbitrarily assigned, before the ratios of the remainder can 

 become determinate. The ratio may be expressed of course in 

 terms of any other weight-coefficient, hence the assumption oik - I 

 ratios is equivalent to assuming k weights absolutely. 



6. Determination of the n - k = m weights. — Reverting to 

 equation (8), the ratio, in terms of which the k — 1 others are 



