414 Mr. A. Cayley on the Geometrical Representation 



whether it may not be necessary to assume, also, that the load 

 upon the luminiferous atoms is a function of the time of oscilla- 

 tion, as well as of the nature of the substance and the position 

 of the axes of oscillation. 



In conclusion, it may be affirmed, that, as a mathematical 

 system, the proposed theory of oscillations round axes represents 

 the laws of all the phenomena which have hitherto been reduced 

 to theoretical principles, as well, at least, as the existing theory 

 of vibrations ; while as a physical hypothesis, it is free from the 

 principal objections to which the hypothesis of vibrations is liable. 



Glasgow, September 2, 1853. 



LXIII. On the Geometrical Representation of an Abelian Inte- 

 gral. By A. Cayley, Esq.* 



THE equation of a surface passing through the curve of in- 

 tersection of the surfaces 



ax* + by* + cz* + dw*=0 

 is of the form 



v(x 2 + y* + z' 2 + w' 2 )+ax 2 + by 2 + cz 2 + dw' 2 = 0, 



where » is an arbitrary parameter. Suppose that the surface 

 touches a given plane, we have for the determination of a a cubic 

 equation the roots of which may be considered as parameters 

 defining the plane in question. Let one of the values of » be 

 considered equal to a given quantity k, the plane touches the 

 surface 



k(a?+t/* + & + ufy+a& + lnf + cs? + dw*=0, 



and the other two values of » may be considered as parameters 

 defining the particular tangent plane, or what is the same thing, 

 determining its point of contact with the surface. 



Or more clearly, thus : — in order to determine the position of 

 a point on the surface 



k{x 2 + y 2 + z' 2 + w 2 )+ax 2 + by i + cz 2 + dw 2 =0 f 



the tangent plane at the point in question is touched by two 

 other surfaces, 



(?(a 8 + y 9 + « 9 + io 9 )+a» 8 + &y a + car 2 + rfii; 8 =0; 

 p and q are the parameters by which the point in question is 



* Communicated by the Author. 



