of a given Surface on another given Surface. 213 



evidently happen whenever h is positive, and the contrary will 

 take place when h is negative ; in which case the representa- 

 tions will have reversed positions. 



In the same manner the representations in 7 and 8 will be 

 similarly or reversedly situated, according as H is positive or 

 negative. 



In order to compare together the representations in 2 and 3, 

 let in the first, d s be the length of an infinitely small line from 

 the point whose co-ordinates are x, y, to another whose co- 

 ordinates are x + dx, y + dy 9 and let / be its inclination to the 

 line of abscissas increasing in the same direction in which we 

 turn from the axis of the x to that of the y, therefore dx = ds 

 cos /, dy =s ds . sin /. In the representation 3, let dcr be the 

 length of the line corresponding to d s, and its inclination to 

 the line of abscissae in the same sense as before, A so that d t 

 = da- . cos A, d u = d or . sin A. We have, therefore, in the 

 notation of the 4th article, 



ds . cos I = da- (a cos A-f «' sin A) 



ds . sin I = d a- (b cos A + 6' sin a), consequently 



, b . cos X 4- b' sin A. 



tang I — . , . . 



P a . cos X + a sin X 



If x and y are now considered as constant, and Z, A as variable 

 quantities, the differentiation gives 



dl __ ab'—a'b . ,,__ ,,. do* 



dk (acos A-|-a'sinA) ,i -)-(6cosX-+-&'sin x) 2 ' '* ds* 



It is therefore evident that, according as ab'—ba' is positive 

 or negative, I and A will increase at the same time, or their va- 

 riations will have contrary signs ; and that in the first case the 

 representations 2 and 3 will be similarly situated, in the other 

 they will be reversed. 



Combining this result with the one found above, we per- 

 ceive that the representations in 1 and 3 will be similarly si- 

 tuated or reversed, according as — is positive or nega- 

 tive. As on the surface, whose equation is ty = 0, we have 

 edx +g dy + kdz = i therefore, likewise (e a +g b + hc) dt 

 + (e a' -\-gb l + he') d u = ; whatever ratio for dt and du may 

 be chosen, the following quantities must be identically zero, 



ea+gb + hc-=. 0, ea'+gb'+hc ! = 



from which it follows that e,g, h must be respectively propor- 

 tional to the quantities bd—cb 1 , caJ—ad, ab'—ba', therefore 



bc'—cb' ca'.—ac ab'—ba 



e g k 



. any one of these three expressions, or if we multiply by the 



quantity 



