394 Mr DAVIES on the Equations of Loci 



For the present we shall confine our attention to the general formulae for 

 the parts of the figure we have been considering. 



XLIII. 



On Contact and Osculation. 



1. In respect to the equation between <f> and Q. 

 Let the spherical polar curves be denoted by 



/(00) = (1.) 



F(dy0-) = (2.) 



Then the values of the radii-vectores when 0, & have received increments 

 and ', will be respectively 



d(f> d*<f) j* , . 



' " 1 " ' 1.2 + 



tfr i "r " I " r " i 



siny'dfl' 1 sin^'dtf* 1.2 



If now for any value of 6 = & we have also < = <p', then sin < = sin 0' : and 

 if we take w = if, our equations (3, 4) will become 



1 







and it will" follow (by reasoning exactly similar to that which is used in the 

 case of differential coefficients of a function referred to rectilinear co-ordi- 

 nates) that the two spirals will have a contact of the order (n) denoted 

 by the number of differential coefficients (n) which are equal in (5) and (6) ; 

 that no spiral which has less than (n) differential coefficients equal to these 

 can pass between (5) and (6) ; that when the order of contact is odd, it is 

 contact only, and when even, that it is both contact and intersection ; and 

 indeed all the general properties which depend upon the coefficients them- 

 selves. I have not, therefore, considered it necessary to put down these ar- 



