Remarks on the " Principle of Least P?-esstire." 197 



function or functions on the determination of which a ques- 

 tion of physics may be made to depend. 



The resultant of the resistances must be equal and opposite 

 to the resultant of the other forces of the system. Now sup- 

 pose that, by the variation of these last forces, the position of 

 their resultant is altered, the position of the resultant of the 

 resistances will then be altered ; but the position of this result- 

 ant cannot be altered unless the resistances themselves be al- 

 tered; the resistances, then, vary with any variation in their 

 position in respect to the resultant of the other forces of the 

 system : they are, therefore, functions of their positions in re- 

 lation to that resultant. 



Or we may reason thus. We must admit the resistances 

 upon different points not symmetrically situated with respect 

 to the given forces to be different, otherwise the resultant of 

 those resistances would have a position determined by the 

 positions of the points, and independent of, and therefore nOt 

 opposite to, the resultant of the other forces of the system. 



Also, the resistances being different from one another, that 

 difference must arise from some cause. Now the nature of 

 the resisting surfaces being supposed to be everywhere the 

 same, or if it be different, being supposed a function of the 

 coordinates of the surface, there is no other cause of dif- 

 ference in the resistances than that resulting from the different 

 positions of the points of resistance. 



Taking, for instance, any one point of resistance ; as long as 

 the coordinates of that point and those of all the other points 

 remain the same, we cannot understand how the pressure upon 

 it should vary; but if we cause the coordinates of that point 

 to vary, we can as little understand how the resistance should 

 remain the same : in fact, it cannot manifestly remain the 

 same ; for if it were, the resultant of the resistances would 

 cease to have its direction opposite to that of the other forces 

 of the system. 



Thus, then, the resistance upon any point remains the same 

 so long as the coordinates of that point remain the same, and 

 it varies when they vary ; it is therefore a function of those 

 coordinates. 



It appears to me that this proof is precisely of the same 

 kind with thai upon which it is usual to assume one variable 

 to be a function of certain others; the proof, for instance, on 

 which we ground the assumption, that if two forces act upon 

 a point, their resultant is a function of the magnitudes of 

 the-'- forces and of the included angle. 



Mr. Kanisliaw's next remark has reference to the case of a 

 mass supported upon three points, forming a triangle, whose 



