APPENDIX. 309 



and also the same curve on another scale, in which y is reduced to one fourth of 



its value, so that 



y = log sin s, 



it is plain that if the second curve is removed parallel to itself by a distance equal 

 to q in the direction of the axis of z, and by a distance equal to logm in the 

 direction of the axis of y, the value of z on the first curve where the two curves 

 intersect each other will be a root of the given equation ; for, since the point of 

 intersection is on the first curve, -its coordinates satisfy the equation, 



y = 4 log sin 3, 

 and because it is on the second curve its coordinates satisfy the equation, 



y -f- log m = log sin (z q) ; 



and by eliminating y from these two equations we return to the original equation, 



m sin 4 2 = sin (z q). 



A diagram constructed on this principle is illustrated by figure 5, and it will 

 be readily seen how, by moving one curve upon the other, according to the 

 changeable values of q and m, the points of intersection will be exhibited, and also 

 the limits at which they become points of osculation. 



On this and all the succeeding diagrams, we may remark, once for all, that 

 two cases are shown, one of which is the preceding example of the planet Ceres, 

 in which the four roots of the equation will correspond in all the figures to the 

 four points of intersection D, D , D&quot;, D&quot; , and the other of which is the very 

 remarkable case that occurred to Dr. GOULD, approaching the two limits of 

 the osculation of the second order, the details of which are given in No. 19 of his 

 Astronomical Journal, and the points of which are marked on all our diagrams 

 G, G , G&quot;, G &quot;. 



2. The second method of representation is by a fixed curve and straight line, 

 as follows. 



(a.) The fundamental equation, developed in its second member, and divided 

 by in cos z, assumes the form 



sin 4 z cos q 



cosz 



= - (tan;? tan q) 



m \ i 



By putting 



x = tan z, b = tan q, a = - 



