74 ' Colorado College Studies. 



But instead of beginning at the origins, we may start from 

 any two corresponding points; the interior tangent circles will 

 be the same. Whence it appears that if « and o.' be any two 

 equal arcs of the circle of the argument, and f^ and /5' the cor- 

 responding arcs of the circle of the amplitude, — /. e., having 

 both extremities of each at points corresponding to the extrem- 

 ities of a and «', — the chords of y5 and ;J' will be tangent to one 

 and the same interior circle. 



From this follows immediately, — at least for the case of 

 the interior system of circles, and it is not difficult to see how 

 this proof might be generalized, — the truth of another theorem 

 of Poncelet, which, in the treatment of that author, precedes 

 the one already quoted; — from which, indeed, he derives that 

 one as a special case. It is as follows: (op. cit., p. 312.) "If 

 three circles, situated in the same plane, have a common secant, 

 real or ideal, and if there be inscribed in one of them a series 

 of triangles ABC whose sides AB, AC touch respectively the 

 two other circles, the third side BC of the triangle will contin- 

 ually remain tangent to a fourth circle, having the same secant 

 with the foregoing three." 



In the preceding paragraphs, the names belonging to the 

 elliptic functions, which are usually defined by the aid of the 

 calculus, have been applied to ratios presenting themselves in a 

 geometrical figure. In order to show that these ratios really 

 sustain such relations to one another as are known to characterize 

 those functions, and hence are entitled to the names, two courses 

 might be pursued. Either differential expressions might be 

 deduced which should lead to the ordinary definitions, or some 

 special property, which, when taken in connection with the rela- 

 tions already given, will distinctively characterize the functions, 

 might be shown to hold true of the ratios in question. I prefer 

 the latter course, and find the property suitable for the purpose 

 in the "addition formula." This is a formula expressing the 

 functions of the sum of two arguments in terms of the functioris 

 of the arguments themselves taken separately. 



