396 ON DIFFERENTIALS. 



coefficients have been used exclusively in the present work, 

 from the conviction that the subject is thus presented in the 

 clearest form, and that if some of the operations are thus 

 rendered a little longer than they would otherwise be, there 

 is at the same time far less liability to error. The equation 

 (2) is certainly of great use in applications of the Differential 

 Calculus, particularly in the higher parts of the Geometry of 

 Three Dimensions: after the remarks already made, the 

 student will probably find little difficulty in those applica- 

 tions. Perhaps he may be further assisted by referring to the 

 theorem for the expansion of a function of three variables. 

 If u = (j> (x, y, z), we have 



</> (x + h, y + k, z + 1) <j> (x, y, z) or Aw 



ax ay az 



where R involves squares and products of h, k, I. Hence the 

 smaller h, k, I, are taken, the smaller is the error contained 

 in the assertion 



. du 7 du 7 du 

 Aw = A -7- + # -3- + * 3-. 



ax ay az 



MISCELLANEOUS EXAMPLES. 

 Find -j- if u = sin" 1 */x ^/(x x*), 



and v = cos" 1 (or cfi] (x% a? x a a 3 ) '-. 



T> U * 



Result. 



Vl x*Jlx*a* 



2. Find the maxima and minima values of (sin #) s!nir . 



3. Find the ar.ea of the greatest isosceles triangle that can 



be inscribed in a given ellipse, the triangle having its 

 vertex coincident with one extremity of the major axis. 



4. APQB is a semicircle whose diameter is AB, and PQ is 



parallel to AB. Draw AQ and BP, and let them meet 

 at R : find the position of P and Q so that the triangle 

 PQR may be a maximum. 



Result. -rj- t must be equal to - . 



