of a given Surface on another given Surface. 1 05 



establish a law by which a determinate point of the second 

 surface is to correspond to every point of the first. This will 

 have been effected if T and U have been made equal to two 

 functions of t and u. These functions will cease to be arbi- 

 trary as soon as they are required to satisfy certain conditions. 

 As X, Y, Z next become likewise functions of t and u 9 these 

 functions must, therefore, besides satisfying the conditions re- 

 quired by the nature of the second surface, also fulfill those of 

 the representation. 



The problem of the Royal Society of Sciences prescribes 

 that the representation shall be similar to the object repre- 

 sented in the smallest parts. It is, therefore, first required to 

 find an analytical expression for this condition. Let us sup- 

 pose that the following equations are the result of the diffe- 

 rentiation of the functions of t and u expressing the values of 

 x,y,z, X,Y,Z. 



dx = adt + aJdu 



dy = bdt 4- b'du 



dz = c dt + c'du 



dX= Adt+ Mdu 



dY = Bdt + B'du 



dZ = Cdt + C'du 

 The condition prescribed requires first that all infinitely small 

 lines proceeding from one point of the first surface and situate 

 in it, shall be proportionate to the corresponding lines on the 

 second surface ; and next, that the former shall form between 

 them the same angles as the latter. 



Such a linear element on the first surface has this expression 

 V(ja z + b 2 + c 2 )dt 2 + 2{aa' + bV + cd)dtdu + (a' 2 + bl 2 -{-c' 2 )du 2 ') 

 and the corresponding one on the second surface is 



\f((A 2 +B 2 +C 2 )dt 2 +2( < AA' + BB' + CO)dtdu+(A' 2 +B'* 



+ C' 2 )^ 9 ). 

 If both are to be in a certain ratio independent of dt and du 9 

 the three quantities 



a 2 + b 2 + c 2 i aa' + bb' + cc\ a' 2 + b' 2 + d 2 

 must evidently be respectively proportional to the three quan- 

 tities A2 + B 2 + Q* A A , + BB , + CC , ? A /2 + B' 9 + C' 2 . 



If we suppose that the values t, u and t + U, u + §u corre- 

 spond to the extreme points of a second element on the first 

 surface, the cosine of the angle formed between the two ele- 

 ments on that surface, will be 



(adt+a f duXah+a^u)^(bdt^b'duXb2t^^u)-^(cdt-\-c'duXch-{-c'iu) 



// [(adt+adu)*+(bdt+b'du)*+{cdt+c'duy~j • [(aST+a'l M )H(*^-|-6'5M) 2 +(c^-|-c'Su)-' n 



New Series, Vol. 4. No. 20. Aug. 1828. P and 



