324 

 fundamental equation, 



v = -. •{ n-F [-, + — + -^, + -T-,+ &c.---—,) . >, 

 n I \p rp r^p r^p t'^'^ pj J 



supposes that the hypothetic thicknesses of the strata of air are 



TT- F 



equal, which is not true, for they vary as . Considering, 



however, that until the law of variation of temperature of the 

 atmosphere between the stations be determinately known 

 (which will, perhaps, never take place), the barometric formula 

 for heights can only be approximate, it is lawful to employ the 

 said formula as closely approximate, until, however, a more 

 correct one be obtained. 



The mathematic error thus noticed escaped Mr. Renny's 

 attention when, six years ago, he gave the formula to Dr. Ap- 

 john. Mr. Renny hopes, at no distant period, to obtain a 

 formula absolutely correct, if not by series, by the integral 

 calculus. 



The Secretary of Council read the following communi- 

 cation from Sir William Rowan Hamilton, on an equation 

 of the ellipsoid. 



" A remark of your's, recently made, respecting the form 

 in which I first gave to the Academy, in December, 1845, an 

 equation of the ellipsoid by quaternions, — namely, that this 

 form involved only one asymptote of the focal hyperbola, — 

 has induced me to examine, simplify, and extend, since I last 

 saw you, some manuscript results of mine on that subject; 

 and the following new form of the equation, which seems to 

 meet your requisitions, may, perhaps, be shewn to the Aca- 

 demy to-night. This new form is the following : 



TV HPJlBl =03_„2_ (i) 



\J{r,-d) ^ ^^ 



" The constant vectors ») and 9 are in the directions of the two 

 asymptotes required ; their symbolic sum, rj + 0, is the vector of 



