traced upon the Surface of the Sphere. 381 



XXXVIII. 



EQUATIONS OF TANGENTS, &c. 



Hitherto our attention has been directed chiefly to those loci which ful- 

 fil certain definite conditions, but which were nevertheless capable of being 

 discussed with very slight appeal to the higher calculus. We shall now 

 proceed, however, to employ the differential and integral calculus for all 

 those purposes in spkcero, which are in any way analogous to the uses which 

 they have fulfilled in the geometry of rectilinear and polar co-ordinates, 

 both in two and three dimensions. The value of any system of conducting 

 an inquiry must be determined by its efficiency and its simplicity combined, 

 and these are qualities which can only be determined by direct experiment. 

 In order, then, to accomplish this purpose in the most effectual manner, we 

 shall deduce a series of formulas for that elementary expression which enters 

 into all these inquiries as a fundamental term I mean the .inclination of 

 the tangent of a curve to the radius-vector in polar curves, and to the axis 

 of co-ordinates in the other case. We shall then be able to find expres- 

 sions for the various lines that can be drawn in specific modes relative to 

 the tangent at a specific point of the curve, in case both of rectangular and 

 polar equations, and the equations of the tangent and the normal in re- 

 ference to every kind of co-ordinate axes that we have discussed. From the 

 results thus furnished, we shall be able to select the most advantageous for 

 the particular inquiries we may be engaged in, as well as deduce some ge- 

 neral principles to guide us in the choice of the co-ordinates which offer the 

 best prospect of becoming successful in any specific class of loci we may 

 wish to investigate. 



We shall commence by means of polar co-ordinates. 



1 . Let f((p 6) = be the polar equation of a spherical curve ; then, if 

 TI be the increment which (the independent variable) receives, the corre- 

 sponding value of (f) becomes, by TAYLOR'S Theorem, 



~ ~ '17273 



3 c 



