244 Mr. Homer on [April, 



its form as any of them, it may be applied to complete the general 

 solution of many equations which have hitherto been solved only 

 in particular cases. For instance, the general value of 



/^ 



dx 



X — \{, .V 



in Mr. Herschell's question, already alluded to, becomes 



log. ^/(a:"— 2cos. 9a;+l)-cot.S x tan.-' ^ - ^°^- - » 



° sin. & 



11. The subject of continued fractions, which has incidentally 

 occurred in this investigation, being intimately connected with 

 the theory of circulating functions, and aifording in consequence 

 another proof of the existence of a sunk fence, so to say, which 

 separates the first and second from the higher orders of func- 

 tional equations, it was my intention to have pursued it in this 

 place. The summation of a finite number of terms of a continued 

 periodical fraction is effected in the Exercises in D. and I. 

 Calculus, only for the two inferior orders, and is not lucidly as 

 to the result, nor even completely, stated in La Grange's Res. 

 des Eqq. Numeriques. But finding that my investigation and 

 remarks would cause too long a digression, 1 reserve them for a 

 future but early opportunity. 



12. Geometry supplies an elegant, and, at the same time, a 

 very simple illustration both of the nature of periodical functions, 

 and of the restriction they are subject to. Setting aside the 

 effect of <p, all our equations are manifestly those of an hyperbola 

 taken between the asymptotes, the excentricity, in equation (32) 

 and the following, being = 2, and the origin of abscissas being 



at the distance — ~ from the centre. Now in our trigonometrical 



a/ a ° - 



solutions, this distance is marked as cot. •&, or cosec. S, or cos. d, 

 to radius 2. Hence, on the conditions of circulating, we have 

 the following construction, regarding ip merely as a mark of an 

 arbitrary addend effecting a proportional and simultaneous 

 transfer both of the line and origin of abscissae. 



With centre C, the centre of a pair of hyperbolas, and radius 

 C X = the excentricity, describe a circle intersecting one of the 

 asymptotes in X, Y, and the other in V, Z. In the semicircle 

 X V Y take any point M which divides it commensurately [say 

 X M : X V Y :: ?/; : ii], and drop the perpendicular M N on 

 X Y. Through X draw the indefinite right line O Q parallel to 

 the common chord of the circle, and of one of the hyperbolas. 

 Through any point A in P Q draw the indefinite right line B D, 

 parallel to X Y, either of the asymptotes. If B D be taken as a 

 line of successive abscissae originating at A, and each equal and 

 similarly affected to the preceding ordinate, then after n opera- 

 tions the series will recommence, and so on in injinitum. 



