NOTICES AND ABSTRACTS 



OF 



MISCELLANEOUS COMMUNICATIONS 

 TO THE SECTIONS. 



MATHEMATICS AND PHYSICS. 



A brief Account of some Researches in the Integral Calculus. By 

 H. F. Talbot, Esq. 



Having been asked to lay before the British Association a notice of 

 my researches in the Integral Calculus, so far as they have been 

 published in the Philosophical Transactions for the present year, I 

 have drawn up a short account of this subject. 



Upwards of one hundred years ago, an Italian geometer, Fagnani, 

 discovered that the difference of two elliptic arcs is in some cases 

 accurately equal to a straight line, whose length is known ; although 

 neither of the arcs, taken separately, can be so expressed. Thus he 

 found, for example, that the quadrant of every ellipse is capable of 

 being so bisected that the difference of the parts is equal to the 

 difference of the major and minor axis of the curve. He also found 

 that the hyperbola possesses similar properties, and also the lemni- 

 scate, and several other curves. He thus with great ingenuity and 

 sagacity opened a new track in the regions of analysis, the existence 

 of which had until his time remained unknown. 



The next considerable step was made by Euler, who showed 



generally that the sura of the two integrals / — ^ + / J^ 



»/ VX ./ V Y 

 may be always rendered equal to a constant, by assuming a proper 

 equation between the variables x and y, provided that X was a poly- 

 nomial in X not exceeding the fourth degree. But if X contained 

 the fifth or higher powers of x, he was unable (except in very special 

 cases) to find any solution of the problem. 



Lagrange, who also endeavoured to remove this difficulty, met 

 with no better success. 



The reason of the failure appears now to be manifest, that the 

 solution of the problem was sought for in the wrong direction. It 

 was attempted always to combine two integrals into an algebraic 



Vol. v.— 1836. b 



