528 MR EDWARD SANG ON FUNCTIONS WITH RECURRING DERIVATIVES. 



the odd from those containing the even powers of the argument. Hence, in 

 general, 



sus nt = ., (e nt + e~ nt \ ; cat nt = r (e nt = e~ nt J . 



13. If we raise the expression for sus t to the n th power, and reply the one- 

 half of the development upon the other half, we find 



2" . sus t" = e nt + e~ nt + ^(e<»- 2 > ( + e®- n A + " ^-=-= (&-** + eP—*\ + etc 

 wherefore 



2 re — 1 sus t n = sus nt + z. susf n — 2 M + ^ — ^ — susf n — 4 k + etc. 



in which expression we have to distinguish the two cases of n even and n odd. 



This counterpart of a celebrated trigonometrical equation is obtained by the 

 very same process, only in this case the operations are all real ; whereas the 

 supposition 2 cos t — x + x~ x used in the trigonometrical investigation is noto- 

 riously contrary to fact and to possibility. 



By treating the function cat t in the same way, we can obtain, according as 

 n is odd or even, the formulas 



n odd 



« / \ 71 71 1 Z' \ 



2 n ~ 1 cat t n = cat nt - =- caU n - 2 jt + —~— catf n - 4 \t - etc. 



n even 



7i / \ 7i n — 1 / \ 



2" — 1 cat t" = sus nt — ^ susf n — 2 jt H- ^ — ^ — susf n— 4 re — etc. 



the last term being halved when n is even. 



My intention at present is not so much to give a detailed treatise on the pro- 

 perties of these functions as to show their close relationship to the sine and cosine 

 of the angular calculus. A striking example of the manner in which the catena- 

 rian supplement the office of the circular functions is seen in their application to 

 equations of the third degree. 



14. All cubic equations may be brought to the form £ 3 =t a z + b — o. and, by 



/ 4a 



making s = x J -y these again can be changed into 



On comparing this last with the four equations 



4« 3 — 3x — cos St = o 

 4x 3 — 3x -j- sin Zt = o 

 4x s — 3x — sus Zt = o 

 4x z + Zx - cat Zt = o 



we observe that if Jf ^-3- ) be equal to any one of the functions of '3t, the value ot 



