MISCELLANEOUS PROPOSITIONS. 411 



Hence since 777, - ~ is the differential coefficient of 

 V (h x) 



n cos" 1 T with respect to x, and ^ - 5-. -r- is the differen- 

 h \ K ~z dx 



tial coefficient of cos" 1 r with respect to x ; it follows by 



fC 



Art. 102 that 



i **? _* 2? xv 



71 COS y = COS r + C7, 



where C denotes some constant quantity. Hence 



z ( ^x J 



T = cos 7i cos T C 

 k V h 



But by hypothesis z must be numerically equal to k when 

 x is equal to h ; and thus (7 must be some multiple of IT ; 



/ C \ 



and therefore cos ( n cos" 1 r - # ) is numerically equal to 

 \ h> / 



/ c\ 

 cos n ( cos" 1 y ] . This gives the required result. 



\ ft/ 



The problem of Arts. 375.. .379 is also solved in Bertrand's 

 Calcul Difftrentiel, pages 512... 519. 



380. We have sometimes to determine the value of -/ 



dx 



from an equation < (a;, y] = 0, when x and y are such that 



dd> (x, y) , dd> (x, if) . 



. y and y -. y/ vanish ; for instance, we have to do so 

 dx dy 



when we are finding the directions of the tangents at a mul- 

 tiple point of a curve. The method of Art. 191 is liable to 

 the objection which is there stated. In Art. 195 another 

 method is given for the case in which x = and y = are the 

 values under consideration. It is easy to make the latter 

 method applicable for any values of x and y ; by a process 

 which is geometrically equivalent to transferring the origin 

 of co-ordinates to the multiple point which may be supposed 

 to be under consideration. 



Suppose that a? = a and y = b are the values to be con- 

 sidered. Put a + h for x, and y + k for y. Then the equa- 



