4 BEPOKT 1867. 



that, by means of the theory of coresolTeuts, we obtain new methods of solving 

 algebraic equations up to the fourth degree inclusive ; and although the above 

 discussion does not embrace equations whose degrees exceed four, it apparently 

 indicates that further residts may spring from the study of non-linear differential 

 resolvents, 



A list of 5500 Prime Kumbers. By W. Bakkett Davis. 



On Finite Solutions of AJf/ebraiccd Equatiotis. 

 By the Eev. Professor li. Haeley, F.E.S. 



On a certain Cyclical SymhoL By the Rev. Professor E. Hakley, F.B.S. 



The object of this paper was to explain the meaning and use of a certain symbol 

 which the author had employed -with advantage in dealing with circular* algebraic 

 functions. Some years ago, while engaged on the theory of quintics, the author 

 found that in the transformation and gener.al treatment of the higher equations 

 circular functions occupy a conspicuous place, and i^lay an all-important part ; and 

 the author was led, by aii attentive consideration of the structure of such functions, 

 to demise a calcidus, whereby operations upon them might be materially abridged. 

 The author had since found that his invention had been to some extent anticipated 

 by Vandermonde, in a Memoir on the Resolution of Equations, published by the 

 French Academy in 1771. The author explained the difference between Vander- 

 monde's process and his O'mi, and showed how he had succeeded lately in enlarging 

 the powers of the latter. Examples were given to illustrate the value of the new 

 symbol, not only as an abridged notation, but also, what was more important, as a 

 working instrument or process. 



On a Theorem in the Integral Calcidus. By Dr. D. Bieeens de Haan. 



The differentiation of an integTal according to a constant mider the sign of inte- 

 gration has been extended by Schlomilch to the case of the limits of the integral 

 depending on this constant. Omitting the correction in case of discontinuity, the 

 formula is 



Now an analogous formula should exist for integration under the integral sign. 

 From (1), when R and r are constants in. regard to p, and so the two last forms 

 vanish, we deduce 



\ dp \ f(p, x)ch^ i dx \ f{p, xyjp, (2) 



Jp J'- Jr Jp 



that is, the theorem for changing the order of integi'ation. In the same manner, 

 from (1) with the two last terms, we deduce, first, 



r^cl^ip,.r)dcc^ dp^^_ '■^^^d.v+^cp{p,Ji)^dp-Lip,ry^^dp + C;. (3) 



and afterwards, after some tran.sformations, . 



/-.7 /iE /iE ^q pq p 



\ f'p| /ip,.r>/.r=j ihyf{p,.r)dp-y 'l^ dp\f{p,V.)dp 

 -J^'^./^(,,R)^.^.-|£^./pJAp,,-)^ 



it'^h 



''^Tp''P' 



{i) 



