My source for this is Rodney Sabine who related it to to me in an email some time ago (I mentioned it on the 'What's New' page but it is worth reproducing here).
First, here is the relevant part of the photo showing (left to right) 'Boss' Tucker, his secretary Mrs Whittington, 'Tus' Sheppard and Mr Leggatt.
What caused this unaccustomed hilarity amongst some senior members of staff, I hear you ask? Rodney takes up the story...
Rodney Sabine wrote:"This is the famous 'laughing' photo. A slight drizzle was falling and Tus Sheppard had put up his umbrella. As the camera started to rotate Tus tried to lower his umbrella but it was stuck. He fought with it and in desperation finally stuffed it down between himself and Mrs Whittington - which is why she's leaning over. If you look at each section, as you move from left to right more and more boys are laughing."
The full photo may be found here - http://www.rgs.saund.co.uk/1964sensection1.html - and, as Rodney says, you can gauge the growth of the laughter as the camera pans from left to right. People in sections 1 and 2 of the photo show barely a glimmer of a smile but from section 3 onwards the gales of merriment grow to a crescendo. You can just make out the umbrella between Tus and Mrs Whittington.