280 Remarlcs on the theory of the Resistance of Fluids. 



Both Prof. W., and the common theory, in order to obtain the 

 value of the resuhs found by them respectively, in the preceeding 

 step, when estimated in the direction of the plane's motion, resolve 

 them after the usual manner of resolving forces. This course on 

 the part of Prof. Wallace is correct ; for the quantity he had deduced, 

 being a force, may be resolved ; but on the part of the common the- 

 ory this course is fallacious, for the quantity it had deduced was the 

 fluent power of resistance, a complex quantity, resulting from the 

 product of two unlike factors. Such a quantity cannot be resolved 

 after the usual manner of resolving forces. It must first be separa- 

 ted into its original elements or factors, and the value of these must 

 b6 estimated separately in the new direction, and not collectively. 

 It is a curious fact that the error in the common theory at this point, 

 exactly balances the fundamental error before pointed out, and in 

 this way makes the ensuing result correct. Whereas Prof. Wallace, 

 on the other hand, by the greater purity of his logic, avoids the sec- 

 ond error, and leaves the first in full force to destroy the truth of the 

 result he deduces. But let us proceed with the investigation. 



We found under the last head, that the force of resistance, in a di- 

 rection perpendicular to the plane, is as the square of the sine of in- 

 clination. This force resolved, gives the force in the direction of the 

 plane's motion, as the cube of the sine of inclination. 



We found under the last head, also, the power of resistance per- 

 pendicular to the plane, to be as the cube of the sine of inclination ; 

 being the product of two factors, viz. oi force, which is as the square 

 of the sine, and distance, which is as the sine. Estimating the value 

 of these two factors in the new direction separately, the force be- 

 comes as before, as the cube of the sine, and the distance becomes 

 a quantity which is given by the hypothesis. Consequently, the 

 power of resistance encountered by the plane in the direction of its 

 motion, in moving a given distance, is as the cube of the sine of in- 

 clination. The same reasoning precisely, mutatis mutandis, applies 

 to the fluent power of resistance, and, therefore, the fluent power of 

 resistance, or the resistance encountered by the plane in a given 

 time, in the direction of its motion, is as the cube of the sine of in- 

 clination. It has been shown, therefore, that the force, power, and 

 fluent power of resistance on the plane in the direction of its motion, 

 are each respectively, as the cube of the sine of inclination. Prof. 

 Wallace makes the first of these, as the square of the sine. The 

 common theory, by the fortunate coincidence of two errors, makes 



