ELIMINATION OF FUNCTIONS. 273 



so that we can replace the right-hand member of (1) by the 

 right-hand member of (2). Hence, as in the preceding Article, 

 we obtain a result which we may write 



T~T I l*2? G/& I / \ I / \ I / \ s\ 



>z> ' ~ ' *' ' ' ...... n = 



Differentiate this again and make use of (1) or of (2) ; thus 

 we obtain another result involving only the same arbitrary 

 quantities. By continuing the process until we introduce the 

 differential coefficients of z of the n th order, we find that we 

 have on the whole n + 2 equations from which we may elimi- 

 nate a and the n arbitrary functions of a. 



253. As an example of the preceding, suppose only one 

 arbitrary function <(). The given equations become 



f{x,y,z, a, </>(a)} = 0, 

 F{x, y, z,a, (a)} = 0. 



Differentiate each with respect to x and y. We thus have 

 six equations, from which we may eliminate 



da dot. . , N 

 *'dx' djj' ^^ 



leaving one equation between 



dz , dz 

 x, y, z, -j- , and -7- . 

 ' dx' dy 



254. The conclusions obtained in Arts. 251, 252 are 

 due to Dr Salmon ; see his Geometry of Three Dimensions, 

 Chapter xn. It had been usual to overestimate . the num- 

 ber of derived equations which are necessary in order to 

 effect the elimination in Art. 252. Suppose, for example, there 

 are two arbitrary functions so that 



F{x,y, z, a, <, fa (a)} = 0, 



then it might appear that by forming the derived equations up 

 to the second order inclusive, as in Art. 248, we should obtain 



T. D. C. T 



