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view to publication, and I selected the most suitable questions which I met with from time to time. Amongst other sources I drew largely from the Tripos Papers of 1875 and 1878, which are generally acknowledged to contain some of the most interestivg of the problems which have appeared of late years, and solutions of these questions have been brought out by the examiners. When I decided to print my collection of Problem Papers, with solutions, two courses were open to me, viz. either to insert the questions selected from these two Triposes, and merely to indicate their origin, or to omit them altogether. Owing to their instructive character I was unwilling to adopt the latte course, and I felt that I should be doing good service both to teachers and to students in bringing these questions before their notice. For the use of those who do not possess the solutions of these two Triposes, I have added in an Appendix an equal number of alternative questions which are to a great extent similar in character to the corresponding problems. I have also added two notes, one on Geometrical Maxima and Minima, and the other on the geometrical method of investigating the envelope of a line which moves subject to certain conditions. The problems bearing upon these two subjects are extremely interesting, although hints as to their treatment are seldom to be met with.

My thanks are due to my former pupils and to my other friends who have rendered me great assistance, but especially to Mr. R. F. DAVIS, M.A., late of Queens' College, Cambridge, who supplied me with many very neat solutions, particularly in geometrical problems. Those marked with an asterisk are all due to him, and it is a source of great regret to me that I was not able to avail myself still further of his aid.

Owing to the nature of the present work it is impossible for me to expect that it is quite free from errors due to transcription. It is hoped, however, that the context will allow them to be corrected without difficulty. At the same time I should be glad to receive notice of any errors, or of suggested improvements in any of the methods employed.

On the following page will be found a list of the errors which occur in the WEEKLY PROBLEM PAPERS, most of which were due to the papers from which they were taken, and were detected on working out the solutions.

JOHN J. MILNE.

CHESTNUT HOUSE,

HEVERSHAM, MILNTHORPE.

ERRATA IN WEEKLY PROBLEM PAPERS,

P. 16, 2nd line, for (-2)

(-2)

P. 4, last line, for do + Ag read do + ajo
P. 9, last line but two, for ag read Az.
P. 15, last line, for a = B read a B.

read
2c

2c PAPER II.

No. 6, instead of the word “in,' substitute 'con

secutive terms of a fixed.' VIII.

7, in the last factor, for + uread a. XII.

2, in first line, for x read xn.

7, in last line, for a sin (a - a') read a sin (a-a). XIV.

4, for cos 70 read cos 50. XVI.

3, log 3 47712. See note at end of solution. XVII.

2, in the last line but one, for x = - , read x XXV.

5, last line, for "intersect' read “intersects.' XXVII. 6, before the word “Shew' insert ‘The paper is

on the point of falling over.'
1

1
XXIX.

for c read c.

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XXXIX.

1

1 1 7, the last line should be +

(aa)2 (63)2 A 2, the first line should be

If p, q, r are all unequal positive integers,

and x is positive and not equal to unity. 6, for 'axes' read 'semi-axes.' 1, for 2nx(1 + x)n-1 read 2nx(1 + x)2n-1.

XLI.
XLII.

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= m = n =

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Pan

2, for

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PAPER XLIV. No. 6, after of 'insert the intersection of.'
XLVII. 1, last line, for read f.

6, third line, for 2a read 2a.
L.
1, in last line, for 1 =

+ read The real values of l, m, no are given by

- 21 LII.

1, for 576 read 2304. LV.

1, fourth line, for "index' read 'radix.'
LVII.

1. (3), for 63 read 61.
Pn?

read
Q2n

Q2n
LVIII.

3, for nt read 2nto.

6, at the end add 'which also touches the chord.' LXI.

6, first line, for “conjugate' read “any two. LXXIII. 5, at the end add 'and if KM and QN intersect

in R', shew that QR' QN.
LXXIV.

1, for 7 read 5.
LXXXVII. 1, first line, for b read a + b.

2, insert 2 in the L of the 2nd fraction.
XCV.
.

1. (3), this series should be
1
2
3

4
12.32

+ + +

32.52 52.72 72.92

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