300 Dr. Lhotsky's Notice of a Mineral Spring 



not either of the signs + must be used, and the question is) 

 which ? 



This will be answered by taking different points in the 

 bounding line, which may be continuous or not. Now every 

 line returning into itself, whether continuous or not, will natu- 

 rally divide with respect of any given system of axes, into 

 at most four parts, or sets of parts; two in which dx and dy 

 both increase or both decrease, and two in which one increases 

 and the other decreases. 



Take P^, P„ P3, P„ any 

 points in the four quadrants 

 respectively, it will be ob- 

 served that. 



At Pj the p column enters additively, and the p' column 

 subtractively. 



At P-, both columns are additive. 



At Pg the p' column is additive and the p column subtrac- 

 tive. 



At P4 both columns enter subtractively. 



Again, reckoning round in the direction of the arrows, 



At Pj, X and y are both increasing. 



At P,2, X is increasing and y decreasing. 



At Pg, X and y both decrease. 



At P4, X is decreasing and y increasing. 



Thus when/(p dy) a\\df{p' dx) are affected with the same 

 signs, dx nx\d dy are of opposite signs; and when^/p r/ 7/, 

 fp' dx are of opposite signs, dx and dy are of the same sign. 



Hence it appears that the mass of the area, whose density 

 at {x,y) is p + p', V is capable of being represented by 



±/>. {(/../.) ^, -(>■..) ^^^}. 



XL VII. Notice of a Mineral Spring, Menero Dowjis, N. S. 

 Wales. By Dr. F. Lhotsky, late of the C. S. Van Diemen's 

 Land.* 



IN the small tract of country as yet known of New Holland, 

 a number of interesting facts relating to mineralogy and 

 * Communicated by the Author. 



