Prof. Encke's Derivation o/'Monge's Formula. 329 



similar to that represented by fig. 11. Hence, if in fig. 12. 

 the 7iorth pole of the magnet were to be placed above the disc, 

 instead of the south pole as there represented, the distribution 

 of force would be indicated by the two systems of arrows in 

 that figure ; the revolution of the disc being in the direction 

 of the large exterior arrow. 



Now, as every condition, both of arrangement and motion, 

 has been considered to be inverted to produce the distribution 

 of force represented by fig. 12, that figure may very well repre- 

 sent the lower side of the plate turned upwards, when the con- 

 ditions of arrangement and motion are represented by fig. H. 

 Indeed it is more convenient to examine the two sides of the 

 disc in this manner; for when the needle is placed below, its 

 motions cannot be very well observed, except at a short di- 

 stance within the edge. 



When the plate is not very large, this force is more equably 

 distributed over the surface, as indicated by the distribution 

 of the arrows in fig. 13 ; bat in no case is it exactly so. 



I have examined the distribution of magnetic polarity in 

 discs and other forms of metallic surfaces with a good deal 

 of attention, whilst the magnetic poles were variously posited 

 with regard to them, and I have collected a number of curious 

 facts; many of which are exceedingly difficult to arrange, on 

 account of the singular windings of the force which actuates 

 the needle. I have, however, succeeded in tracing the dis- 

 tribution in some instances by experiments, which will be de- 

 scribed in an early communication. 



Errata and Corrigenda in the former portion of this paper. 

 Fig. 3. of the Plate is illustrative of Experiment 3, page 273. 

 Page 274, line 22: for bda read da. 

 Transpose the letters S N in Fig. 5, 6, and 8 of the Plate. 



XLV. Derivation of Monge' a Formula; for the Transformation 

 of Coordinates in Space. By Professor Encke*. 



IN the transition from one system of rectangular coordinates 

 in space x, y, z to another likewise of rectangular coordi- 

 nates .r',y,:;', the latter, being functions of the former, are 

 found by these formulae, the point of beginning of the two 

 systems being supposed the same : 



x' = ax + dy + a" 2 

 y^ = bx + b<y + V'z 

 z' = cx+dy+d'z 



The coefficients a, b, c, &c. are the cosines of the angles 

 which the axes of the coordinates respectively form together. 



• From Encke's Ephemeris for 1832, p. ."iO.!. 

 N.S. Vol.11. No, 65. 3% 1832. 2U If 



