380 



ginedjand he points out several precautions which should be observed 

 in placing compasses on board such vessels. 



' Researches on the Integral Calculus. Part I." By Henry Fox 

 Talbot, Esq., F.R.S. 



The author premises a brief historical sketch of the progress of 

 discovery in this branch of analytical science. He observes that the 

 first inventors of the integral calculus obtained the exact integration 

 of a certain number of formulas only j resolving them into a finite 

 number of terms, involving algebraic, circular, or logarithmic quan- 

 tities, and developing the integrals of others into infinite series. The 

 first great improvement in this department of analysis -was made by 

 Fagnani, about the year 1714, by the discovery of a method of rec- 

 tifying the differences of two arcs of a given biquadratic parabola, 

 whose equation is x 4 = y. He published, subsequently, a variety of 

 important theorems respecting the division into equal parts of the 

 arcs of the lemniscate, and respecting the ellipse and hyperbola ; m 

 both of which he showed how two arcs may be determined, of which 

 the difference is a known straight line. Further discoveries in the 

 algebraic integration of differential equations of the fourth degree 

 were made by Euler ; and the inquiry was greatly extended by Le- 

 gendre, who examined and classified the properties of elliptic inte- 

 grals, and presented the results of his researches in a luminous and 

 well-arranged theory. In the year 1S28, Mr. Abel, of Christiana, in 

 Norway, published a remarkable theorem, which gives the sum of a 

 series of integrals of a more general form, and extending to higher 

 powers than those in Euler's theorem ; and furnishes a multitude of 

 solutions for each particular case of the problem. Legendre, though 

 at an advanced age, devoted a large portion of time to the verifica- 

 tion of this important theorem, the truth of which he established upon 

 the basis of the most rigorous demonstration. M. Poisson has, in a 

 recent memoir, considered various forms of integrals which are not 

 comprehended in Abel's formula. 



The problem, to the solution of which the author has devoted the 

 present paper, is of a more general nature than that of Abel. The 

 integrals, to which the theorem of the latter refers, are those com- 



prised in the general expression^ where P and R are entire po- 

 lynomials in x. Next in order of succession to these, there naturally 

 presents itself the class of integrals whose general expression is 



/*P d x .... 

 f-r-_ - — , where the polynomial R is affected with a cubic, instead 



J, \/R 



of a quadratic radical; but Abel's theorem has no reference to these, 

 and consequently affords no assistance in their solution. The same 

 may be said of every succeeding class of integrals affected with roots 

 of higher powers. Still less does the theorem enable us to find the 



sum of such integrals as J*<p (R) dx; R being, as before, any entire 



polynomial (that is, containing at least two different powers of $), 



