VOLUMES OF SOLIDS AS KELATED TO TRANSVERSE SECTIONS. 39 



sufficient condition for this is, that each quantity included within 

 brackets on the left-hand side of (4) must be equal to the corres- 

 ponding quantity within brackets on the right hand side; that is 

 to the quantity which is to be multiplied by the same constant. Or 



«a« + j86- +7 «*+ etc. ^a + ff + 7 + etc. (g) 



r ' u+ L 



where u denotes p, q, or r, etc. This last equation may be written, 



a(a«- _J_\ + /3(&«- _i\ + 7 ( c »_ _J_) +e tc. = (6). 



v u+1 / r u+1 / /v u+1' v/ 



It will therefore be requisite to solve merely a system of equations 



of this last form, viz. (6), the qucesitum being a, f3, etc., when the 



indices are supposed known. Put 



<h = a?-— — - ; b x = b*- — — ; etc. 1 



? t P V < 7 > 



a 2 = a <l - ; b 2 = 6* ; etc. 



#+1 q+1 J 



and similarly throughout : then the system of equations of the 



type in question, viz. (6) is simply the linear system 



a x a + b-fi + . . . + w x v == ^. 



a 2 a + b 2 /3 + ..+n 2 v = I , g 



Um a + b m /3 + ... + n m v = 

 there being n terms in each of the m lines or rows ; or, disregard- 

 ing the zeros, say n columns and m lines ; that is an assemblage 

 of m linear homogeneous equations in n variables. Hence if 

 n = m+l, then although there will be a one-fold infinity of 

 solutions, the ratios of a, /?, y, etc., will be determinate -, 1 and 

 since these represent relative weights, they are all that are needed. 

 Consequently, 



Prop. (a). In order that the ratios of the weight-coefficients \ 

 positive or negative, of values of the original junction, arbitrarily 

 taken for any different values of the variable, shall be determinate, 

 it is necessary that, in general, the arbitrary values be m + 1 in 

 number, m being the number of different indices in the function. 



1 It is obvious that in (8) any common multiplier of all the terms will 

 not vitiate any equation, hence the relative value of the variables is all 

 that such a system can furnish. 



