290 



Mr DAVIES on the Equations of Loci 



Or putting in (8) the values of sin \ cos \ from (6, 7), we have an equa- 

 tion of the form 



A* + B 2 cos 2 /c + 2 BC cos K sin K + C 2 sin 2 /c 

 D 2 cos 2 /c 2 DE cos K sin K E* sin 2 K 



Where A = cos e y cos a, 



B = cos e, sin a, cos /3, 



C = cos , sin a, sin /?, 



D = cos A, sin a, cos /3, cos a y sin A, cos /<", 



E = cos \ sin a, sin /3, cos a, sin A, sin K/ 



= (9.) 



.(10.) 



r 



D z ) B 2 D 2 C 2 E 2 



--- " 



DE-BC --- DE-BC " 



G H sin */c = 2 sin K cos /c ; 



Or squaring, transposing, &c. 

 G 2 



. 



(11.) 



G [ 





c 



Or putting for G and H their values, we have 



s . in2K _(A 2 +B 2 -D 2 )(B^^ 2 "-C^E r2 )+2(DE-BC) 2 2(DE-BC)\/(DE-BC) 2 -(A 2 +C 2 -E 2 )(B 2 -D 2 -C a -B 2 ) 



(B 2 D 2 C 2 E 2 ) 2 + 4 (DE BC) 2 



cos=7c=- 



(B* D 2 C 1 ^TE 2 ) 2 + 4 (DE BC) 2 



These expressions in the general solution, and in their present form, are 

 very complex. If, however, facility of calculation be aimed at, they may 

 be simplified in various ways. But as my object in this paper is a totally 

 different one, that of obtaining analytical formulae to express the constants 

 of an indeterminate geometrical problem, which are only implicitly given, 

 it is unnecessary to dwell upon the question in any other point of view. In- 

 stead, therefore, of attempting any transformation, I shall conclude the 

 article by a remark on the signification of the solution just obtained. 





