.Oa the Mechantfin ef the Eyi. r^^ 



w to « ; take r ~ \ —\ , and let the denfity be fuppofed to vary every where in- 



,:. ,;. 1^;.;:;;-!; ^ log. w — log. « '-i-o-k: -i: '■- .-• ■ 



verfely as the power — of the diftance from the centre : then the principal focal diftante 



■from the centre will be ■ . —r—. — When r=: i, it becomes— -r—r, ~ •• 



2 nl — m 2(H. L, »» — H.L. «) 



For a lens, deduft one-fourth of the difference between its axis and the diameter of the 



fphere of which its furfaces are portions. 



CofoUary. If the denfity be fuppofed to vary fuddenly at the furface, m mud exprefs 



ihe difference of the refraftions at the centre and at the furface ; and the focal diftance, 



thus determined, rauft be diminiflied according to the rcfradtion at the furface. 



Propofithn VIII. Froblem. 



To find the nearer focus of parallel rays falling obliquely on a fphere of variable 

 denfity. 



Sslutian. Let r be as in the laft propofition, s the fine of incidence, t the cofine, and e 



the diftance of the focus from the point of iminerfion. THeh' is* =i ■> ^ ■,w beings 



r + i 



(aA + l>li + cC_+.. .) +2a A + 6*B/- + lo <r C .r* + 



'''-*- 1 i.ii hju!ii)rTO> ow}"jn':'(i3 ; 



(r-.i> 



wher« a = — ^', i = — ^^-— , c = — , A = i, B = 4^ A^ C ss-^Bi D = I'O. 



r + 1 2 r — I S ^—3 ■ 



Bat, when / is large, the latter part of the feries converges fomewhat flowly. tte former 

 part might be abridged if it were neceffary ; but, fince the focus in this cafe is always very 

 imperfe£V, it is of the lefscqnfequence to provide an eafy calculation. 



Gemral Scholium, Thp two firft propofition's relate to well known phenomena; the 

 third can hardly be new } the fourth, approaches the neareft to Maclaurin's conftruQion'', 

 but is far more fimple and convenient ; the fifth and (ixth have no difficulty ; but the two 

 laft require a long demonftration. The one is abridged by a property of logarithms ; the 

 other is derived from the laws of centripetal forces, on the fuppofition of velocities direflly 

 as the refractive denfities, correfting the feries for the place of the apfis, and making th6 

 Cue of incidence variable, to determine the fluxion of .the angle of deviation. 



v.. Dr. Porferfield has employed jjn.experitnent, firft made by Scheiner, to the' dftcr- 

 mination of the focal diftance of the eyej and has decrlbed, under the name <>ran 

 optometer, a very excellent inftrument, founde^ on the principle of the phenomenshi. * 



# Ed»b,M<d,M. ypi. iyl_y. wi. 



■ Ll2 B'^' 



