354 Mr DA VIES on the Equations of Loci 



NOTES. 



NOTE A. 



IT will often facilitate our detection of circular loci, to frame a table of solutions 

 of (I. 1.), with respect to each of the functions of the arcs employed. The follow- 

 ing is a specimen : 



,. Great Circle, or = 5. 



Less Circle to radius j. 2 



_ con sin X cos ~*. -+- cos X Vsin 2 j sin'Xsin'd x. _ H^cotX _ -t-cotX 



1 _ sin* X sin 2 6 *. Vl_ginX sin*<P-x ~~ Vcot* X -f- cos* 0~H 



cos? cosX;+;sinXcos() xvsin 2 ^ sin 8 Xsin 2 x, ;4;cos0 x. sin X 



cos p = 



1 sin s Xsm*0 x. *!\ sin ! Xsins0 x. Vcot'O. 



cos f cos X -H sin X cos 6 </sin 2 e sin* X sin* x. . cos x. -. 



cota=- ~ . , - = -., =tanXcos0 x 



cos sin X cos 9 x. -+- cos X^/sin* g sm X sin^ x. 



The tangent, secant, and cosecant, being the reciprocals of these, may be formed 

 by inspection, and need not be tabulated. We might have inserted the other func- 

 tions of 8, or we might have resolved the primary equation in respect of 6, and in 

 terms of each function of <p ; but the process is too simple and elementary to need 

 more than pointing out to the reader. 



NOTE B. 



It becomes of great importance to form an analytical table of. the significations 

 of spherical equations; that is, for determining the points signified by^ptf) 0, re- 

 solved for either of the variables. We propose to furnish one in this note. 



All trigonometrical functions belong to more than one arc to innumerable arcs. 

 So long, however, as we attend only to the points in which those arcs terminate, 

 these points are, for each function, confined to two, the successive pairs terminating 

 at the same points after the increment of an indefinite number of complete circum- 



