CHAP. XVIII.] ELEMENTARY ILLUSTRATIONS. 293 



10. Now a difficulty, the bringing of which prominently be- 

 fore the reader has been one object of this investigation, here 

 arises. How shall it be determined, which root of the above 

 equation ought to taken for the value of X. To this difficulty 

 some reference was made in the opening of the present chapter, 

 and it was intimated that its fuller consideration was reserved for 

 the next one ; from which the following results are taken. 



In order that the data of the problem may be derived from 

 a possible experience, the quantities p, q, and r must be subject 

 to the following conditions : 



1 +p-q-r >0, 



l+q-p-r>Q, (14) 



l+r-p-q>0. 



Moreover, the value of X to be employed in the general solution 

 must satisfy the following conditions : 



\>-, - - - , X>. - l , X^ 1 - - - . (15) 

 1+p-q-r l+q-p-r l+r-p-q 



Now these two sets of conditions suffice for the limitation of 

 the general solution. It may be shown, that the central equation 

 (13) furnishes but one value of X, which does satisfy these con- 

 ditions, and that value of X is the one required. 



Let 1 + p - q - r be the least of the three coefficients of X 



given above, then - -- will be the greatest of those va- 



lues, above which we are % to show that there exists but one value 

 of X. Let us write (13) in the form 



-4{(p + ?+r-l)X + l) = 0; (16) 



and represent the first member by F. 



Assume X = - -- , then V becomes 

 1 + p - q - r 



which is negative. 



