WHEN INDETERMINATE IN FORM. 169 



the term ( j- )- 7 -^f . This term would not occur if -f- were 

 \dyj ax ax 



d*y 

 a constant quantity, for then -jjr would be zero. Hence 



equation (2) of Art. 190 may be derived by differentiating 

 the equation 



Sdu\ + fdu\ dy = Q 



\dxJ \dy) dx 



with respect to x and treating -^ as if it were a constant. 



Similarly, equation (2) of Art. 191 may be deduced from 

 equation (2) of Art. 190 by differentiating with respect to x 



and treating - as if it were a constant. 



194 If in equation (2) of Art. 190 we have f-p J = 0, 

 then 



dy__ 

 ~ 



dx~ / d*u \ 

 \dx dyJ 



as one value of /-. The other value of -?- will be infinite, 

 ax ax 



for we know from Algebra that if we have a quadratic 

 equation and the coefficient of the highest power of the un- 

 known quantity gradually diminishes without limit, then 

 one of the roots simultaneously increases without limit. See 

 Algebra, Chapter xxil. 



195. The value of ~ , when the values x = 0, y 0, make 

 it assume an indeterminate form, may often be more simply 

 found thus. We have only to seek the limit of - as x and y 

 diminish without limit ; this is obvious from the meaning of 

 -^ , or from Art. 145 ; it will be seen too if we refer to the 



geometrical illustration of Art. 38. 



Example. y* + 3a?y* katxy a*x* = 0. 



