1824.] Mr. Hcrapath on the Solution qf^ n x = x. 327 



multiple of X, for that would give a VT =± i - JL ~~— v a <> a vame 

 which could never become = by any finite value to v, unless 

 at the same time b r = — c r ; and, therefore, the function could 

 never be periodic. Fourthly, z must be such that when v = n, 

 the order of the periodic function, sin v z must be = o. Assume 



therefore z == — , and it is evident the first condition is satisfied 



n 



by k being any independent variable ; the second by its never 



13 5 



becoming a -, -, -, &c. multiple of n, and the third by its never 



being a 0, 1, 2, 3, &c. multiple of the same index unless simulta- 

 neously b r = — c T . Finally, the fourth case is satisfied by k 

 having integral values only ; for otherwise k varying indepen- 

 dently of v could not generally give sin , and, therefore, a zr 



= o when v = n. 



Moreover it may be further observed, since by (13), (14), 

 a r d t = a t d,., and consequently b T+t — b t b r = c r +, — c, c r , that 

 k must be the same in a VT as in d vr , and the same in b vr as in c„ r ; 

 but not necessarily the same in a vr or d rr as in b rr or c„ T . This, 

 therefore, gives the number of functional roots n- for the same 

 form off, and is another instance of deviation from the algebraic 

 analogy. It arises from a similar cause to the preceding devia- 

 tion, an anticipation in part, as it will presently appear, of the 

 arbitrary function in the constants a r , b r , c T . 



For the sake of brevity, we shall adopt one common indeter- 

 minate integer which will enable us to give our final result some- 

 thing simpler. 



V T x = <p-' 



vkx f k X . vkx ' 



sin — \ (b — c r ) cos — .sin — 



n b T + c y ) vkx n n 



a r + — ■< cos — + ; > f x 



. kX k X \ n . k X 

 sin — 2 cos — / (& P + e r ) sin — 

 n n L n ) i (21) 



~ , " kX . vkX\ I 2 2kX A . vkK J * 



V (b T — c,.)cos — .sin — / ( J^-2A,.c,.cos — +c r a sin — 



b r + cr 1 vk>. v n n \ \ n In 



. < cos > — : — <*> x 



kX 1 n . k X I I 2kX \ . kX \ 



2cos T £ ( 6i . + rr)sl n— . ^ 2flr ^cos — + ijsin — j 



This formula is in many instances better adapted for practice and printing under the following form , 



V r x = $ - ' 



sin 



vkX kX „ f . vkX . kX ., . kX . vkh) "1 



— . cos — .2(i,+ ■u*>- + c >) cos — .sin — + {b r — c,.) cos — . sin — ( <f> .r ' 



n n ( n n n n > I 



VkX \ . vkx kX 



vkh . k X ,. kX vkx 



I IkX \ . vkx kX 



bf— 2 b r c r cos — + r,. 4 Ism — .cos — > 



(» r + c r )cos -.sm- -(» r -c r ) cos-, sm- -^— !* | 



a,.(cos- + l) j 



lich is produced from the other by merely multiplying the terms of the numerator and denominator by 



k X kX 



■in — .cos — . 



n n 



