Application of Criteria of Integr ability. 95 



criterion of integrability of a differential expression of two va- 

 riables. Besides the variables x and y, P and Q may contain 

 quantities unaffected by the sign of differentiation, and depend 

 for their values on the values assigned to these quantities. 

 Let us suppose as a particular case that P = and Q = 0, 



we shall then have -j— = and -= — = 0, and the equation 



dP dQ . ... .„,r™ -t 



-j— = -j- ; is numerically satisfied, lne question I propose 



to consider is, whether because this equation is thus satisfied, 

 V dx + Q dy is necessarily an exact differential when P = 

 and Q = 0. 



Very simple considerations will enable us to answer this 

 question. We have seen above that the only condition of in- 

 tegrability is, that -z — and -j— ' be identical quantities. In 

 every case, therefore, the ratio of these quantities must be a 

 ratio of equality. But when -v— = and - f — = 0, that ratio 



is expressed by a vanishing fraction — , which is not necessa- 

 rily equal to unity. Consequently P d x + Q dy is not ne- 

 cessarily an exact differential when P = and Q = 0. 



The above reasoning readily suggests the process to be fol- 

 lowed for determining in any particular instance in which 

 P = and Q = 0, whether P dx -f Q dy may be regarded as 



an exact differential. Let the given expression be a (e — 1 )ydx 

 + btxdy. Then P = a (/ - l)y, Q = btx, Sl-i «(/-l), 



dQ . . . , . a d? A dQ. ale*- 1) 1Pj n 



-5-2 = bt, and the ratio of -r— to ~ is - v . — '-. If t = 0, 

 dx dy dx bt 



each of the quantities P, Q, — r-, -7- vanishes, and the ratio 



1 dy dx 



of -y— to -j — becomes — . But the real value obtained by 

 dy dx ' 



the usual rule for finding the values of vanishing fractions is 

 -T-. As this quantity is not equal to unity unless a = b, the 



expression a (e — \)y d x + btxdy cannot be considered an 

 exact differential when t — 0, unless a — b. 



Analogous reasoning may be employed to show that P dx 

 + Qdy + 1Hdz\s not necessarily an exact differential when 

 P = 0, Q = 0, and R = 0. 



