It is usual to begin a sermon with a text and if I were making a sermon to-night my text would be from a favourite poet who, unconsciously and unsuspectingly, I fear, described the ideal actuarial department when its total staff is three and by implication described the ideal methods of examining actuarial problems.

Probably all of you have played cricket and football, and if I were to say that you could learn those games by reading about them in books you would laugh me to scorn. The furthest you would go would be to admit that having played a game to some extent you might get useful hints from books or conversation. Some of you might even feel that in a few cases the hints do as much harm as good by tending to reduce originality which, after all, counts for a good deal in any game, and by introducing an element of surprise may in games, as in war, win when a stereotyped method is doomed to failure. It is a steep descent from these high altitudes to the subject we are discussing tonight, but I suggest that what is true of games is also true, to a large extent, of graduation. You will never learn how to graduate by reading about it. Take an experience, that is, the exposed to risk and the deaths (or the week's sickness or whatever the subject implies) or the rates of mortality, and see what they look like and appreciate the hideous unevenness and spend time in trying out of your own head to ‘make the rough places smooth,’ and after that begin your reading.

It is proposed to deal in somewhat greater detail than is customary with the reasons why we are entitled to separate symbols of operation from the quantities upon which they operate, combine them together as if they were ordinary symbols of quantity and interpret the results in terms of other operations.

As a preliminary it may be well to say a word or two on the three primary laws of algebra, the law of distribution, of commutation and the index law and what these imply. Chrystal, in the first chapter of his Algebra, starts from the primitive notion of numbers (i.e. positive integers) and shows that the essential laws according to which these whole numbers combine are these three primary laws. He then shows how the meanings and properties of negative and fractional numbers are fixed by the condition that the meanings which are given to them are to be such as will give consistent results when we combine such numbers with other numbers according to the three primary laws. An illustration from the index law will perhaps make the point clearer.

Probability questions are particularly difficult to the student, since the methods of solution are varied, and, unless the would-be solver has considerable experience, the best method of attack is not easily detected. The only way in which a satisfactory grasp of the subject can be attained is by working on a variety of problems of different kinds. The student will then find that questions in probability tend to fall into certain groups, but, even when he has arrived at this conclusion, he will discover, also, that the boundaries of these groups are by no means well defined, that a solution may often be reached by quite different lines of reasoning and that, except in the simplest cases, it will need all his experience to make a proper classification of any particular example.

The object of this note is to assist students to work on these lines by affording him examples of various kinds—with hints for their solution. We have deliberately refrained from giving the solutions in extenso, in order to encourage the student to work the examples through and sd obtain the practical experience which is so essential to proper knowledge.