Mr. A. Cayley on the Integral f Ax -r- \/ (x + a) (x + b) (x + c) J 281 



their course in all countries under ordinary conditions, wlien no 

 subsidence or elevation is occurring, was traced by Hutton. 



Even if, in ancient periods, the rate of denudation was greater 

 than at present, and the supplies of detritus to rivers more ex- 

 tensive, the fluctuations of the sea-level and the elevation of the 

 beds and plains of rivers would have been proportionately greater. 

 There would, therefore, still have existed some localities where 

 the rate of the formation of alluvial plains near the sea kept pace 

 with the elevation of the waters ; so that, as at the present time, 

 conditions would have existed for the accumulation of fluviatile 

 strata containing terrestrial remains without any subsidence of 

 the land. This is a subject, however, that must be further 

 studied, more especially when its value is considered in relation 

 to the great masses of fluviatile strata either of the Mississippi, 

 the Ganges, the Nile, or the Po. For the above reasons it would be 

 difficult to determine, when examining sections of thick fluviatile 

 strata, whether these accumulations of detrital matter had been 

 formed during subsidence of the land, or during the gradual 

 elevation of the level of rivers and seas, arising from the con- 

 tinual operation of ordinary physical causes. 



XLII. Note on the Geometrical Representation of the Integral 

 /dx-i- // (x + aj(x + b)(x + c).l By Arthur Cayley*. 



THE equation of a conic passing through the points of inter- 

 section of the conies 



ax^ + by^ + c-s-^ = 

 is of the form 



w{x^ + 2/H z^) + ax^ + by'^ + cz^=0, 

 where iv is an arbitrary parameter. Suppose that the conic 

 touches a given line, we have for the determination of z^ a qua- 

 dratic equation, the roots of which may be considered as para- 

 meters for determining the line in question. Let one of the 

 values of w be considered as equal to a constant quantity k, the 

 line is always a tangent to the conic 



k.{x'^-\'y'^ + z'')-^ax^ + by^-\-cz^=:0', 

 and taking w=-p for the other value of w, p is a parameter 

 determining the particular tangent, or, what is the same thing, 

 determining the point of contact of this tangent. 

 The equation of the tangent is easily seen to be 



^ s^b — c \^a + k Va-\-p-\-y \^c—a WVk \/b^-p 



+ z Va — b Vc-\-k Vc-\-p = 0. 



* Communicated by the Author. 

 Phil Mag, S. 4. Vol. 5. No. 32. April 1853. U 



