$0 Mr. T. S. Davies on the Foci and Directrices 
Again, from (1, 3, 8, 9.) we have, after slight reduction by 
means of the relation = 2/ ac, the equations 
Va.cosa—V¢C 
SS os ae ee . . . . 1 . 
sin? « em) 
V/e.cosa—V a « «, Cee) 
sin? ce 
a 
Ae 
the sameness or difference of the signs of which are to be de- 
termined by the relation (2.). 
From (4, 5.) and (9.), we have 
2 (i + h'cosa) =—(d'+2pr'sin®a) . + - (12.) 
2u2 (hi + cosa) =—(e + 2g7'sin?a%) « « - (13.) 
Resolving which for 7’, k!, and putting Q for w? sin? a, 
d'—e +2(pcose—q) 7 sina 
i 
— 2Q » (14) 
, _¢ —d' + 2(qcos 4 — p) 7” sin?® @ 
I! MaIMASacagieile (* silal lial aRieaanT A a (15.) 
Now we have from (10, L1.), 
Va.cos?a—V c.cosa+V7e.cosa—V rs 
ps ear Os? ae c.cosa—Va _ _ yy 16.) 
Vc.cos2«—Va.cosa+V a.cosa—V ¢ as 
Ge ss ni dasinl, Uninet el ae VC (17). 
Insert (16.) in (14.), and (17.) in (15.); then we get 
d'——2 Va.r'sin?a 
i= Ones: . (18.) 
d—e— 2 c.7'sin?a 
ki =— zQ ‘ - (19.) 
Express by means of (18, 19.) the value of h” +2 2’ k' cos a 
+h, and insert in (6.); then after very slight reduction, 
d'—e' 
~ 2(Va+%Vc)sin?a * 
Insert the value of 27! sin? from (20.) in (18, 19.), and 
there results 
# ag ee (BBD 
_ (d-e)Ve 
i= 2Q(VatV oC) oe: Set Me™ of) Tee (21.) 
Fea (d’—e)V a (22) 
SOV a +i thos nis 
We have hence only to correct these results for the coordi- 
