376 Prof. Magnus on the Constitution of the Sun. 



It is also applicable to explicit integration, and teaches us that 

 if any function of x is integrable with regard to x> its inverse is 

 not only integrable, but its integral is known without any new 

 process of integration. This is implied in the identity 



J''^r- 1 xdx = xyjr- 1 x — 'ijr l '^r- 1 x; -^r x t being / 'tytdt. 



Thus, if 



/*cos xdx= sin as + c, y*cos -1 xdx—x cos" l x — sin cos'^x+c. 

 If 



J*e x dx = e*, y* log xdx = x log x — e lo & * + c. 

 If 



ft&nxdx = c— log cos a?, y , tan- 1 #cfo?=#tan -] a? 



+ c — log cos (tan~ 1 x) ; 

 and so on. 



Even if the inverted function be not expressible, yet we can 

 express its relation with its integral. Thus if Xx denote the 

 inverse of xe* 3 smcef xe x =xe x — e*, we have 



jXxdx = x\x -f- e Xx — \xe^ x 4- c. 



liXx denote the inverse of ax n + x, then (using an obvious re- 

 duction) 



J' 



^7 n -v In — l/^\o 



Kxdx = r- xxx — =- [KxY + c. 



n+l 2/i+l v ; 



I do not remember seeing this method given in elementary 

 works as a process of integration. 



Fitzwilliam Square, Dublin, 

 April 5, 1864. 



LIX. Note on the Constitution of the Sun. 

 By G. Magnus*. 



SO long ago as the year 1795, W. Herschel put forward the 

 view that the sun consists of a dark nucleus which is sur- 

 rounded with a photosphere, or atmosphere giving out light and 

 heatf. Between the latter and the nucleus he assumed the 

 existence of a light-reflecting atmosphere, which, in virtue of 

 this property of reflecting, hindered the illumination of the 

 nucleus by the photosphere. In treating of this hypothesis J, 

 which he designated as the one generally adopted §, Arago made 



* Translated by Prof. Wanklvn from Poggendorff's Annalen, No. 3, 1864. 



t Phil. Trans. 1795, p. 46. 



% Astronornie, vol. ii. p. 94 (Arago). § Ibid. p. 143. 



